Transcript Basic Image Processing

Basic Image Processing
February 1
Homework
Second homework set will be online Friday (2/2).
First programming assignment will be online Monday (2/5).
Slides will be posted online before the end of next Monday.
Today, we will finish Chapter 6. We will work on Chapter 7 and
8 next Tuesday.
Recall that we introduced a linear noise model
Optimal filtering: Find a filter that maximally suppresses
the noise.
An application of the math we have been studied so far!
Changing the notation slightly (following the textbook)
o is the output signal, with h the (unknown) filter
kernel (point-spread function) that we want to figure
out.
Find h that minimizes this error functional.
The expression for E become complicated (only in appearance !)
Auto-correlation and cross-correlation
Given two functions, a, b, their cross-correlation is
defined as
The auto-correlation of a function is the cross-correlation
between itself:
Recall that cross-correlation and auto-correlation are
functions, not just a number.
Some important properties of correlations.
(0, 0) is a global maximum of
auto-correlation.
If
then
The Fourier transform of autocorrelation is called power
spectrum.
The cross-correlation of the two images have maximum at
(100, 200).
The two auto-correlations are the same (invariant under
translation).
Back to optimal filter design
and
Remember, we want to determine h. This is a calculus of
variation problem!
Want to find h such that
That is,
How to get h?
That is, we only need to know the power spectra.
From
Assume that the noise and signal are not correlated.
We have
is the signal-to-noise ratio (for each frequency).
SNR is high, the gain is almost unity.
When SNR is low (mostly noise), the gain is small.
Similar applications
Suppose now we have
Examples:
Image Blurring:
Defocusing:
Motion Smear: image points smeared into a line.
Need to figure out
Assume that the noise and signal are not correlated.
SNR is high
In parts where SNR is low
The gain is roughly